Regenerators are periodic-mass-flow heat exchangers in which a fluid is periodically pumped back and forth through a matrix. During one part of a flow cycle, the matrix absorbs heat from the fluid, and when flow is reversed, heat is transferred from the matrix to the fluid. Two key factors in operation of these devices are the heat exchange between the fluid and the matrix and the heat storage capacity of the matrix. These factors can be characterized by numerical coefficients as follows:
(a) heat exchange-hA PA1 (b) heat storage-C PA1 a. for a given heat exchange, the pressure drop and void volume should be minimized; or, conversely, for a given pressure drop and void volume, the heat exchange should be maximized. PA1 b. the matrix heat capacity must be large enough to keep the temperature swing during a blow period to a small value. PA1 (1) crossed rod or wire screens, PA1 (2) randomly packed sphere beds, and PA1 (3) perforated plates. PA1 (1) perforated plates versus sphere beds for equal pressure drops and identical regenerator dimensions, PA1 (2) perforated plates versus screens for equal pressure drops and identical regenerator dimensions, and PA1 (3) perforated plates versus screens for equal pressure drops, equal regenerator void volumes, and equal regenerator lengths. PA1 (1) perforated plates provide at least a sixfold improvement in performance over packed sphere beds, PA1 (2) for equal regenerator volume, perforated plates are better than wire mesh screens for some ranges of Reynolds numbers, and PA1 (3) for equal void volume, perforated plates are superior to wire mesh screens at all Reynolds numbers.
where h is the heat transfer coefficient (SI units-watts/m.sup.2.K), A is the heat transfer area (SI units-m.sup.2), and the C is the matrix heat capacity (SI units-Joules/kg.K).
There are two additional secondary factors for regenerator operation, namely, pressure drop (.DELTA.P) (SI units for pressure-Pa) across the regenerator due to frictional losses and void volume (VV) (SI units for void volume-m.sup.3) of the regenerator. The pressure drop must be overcome in order to drive the fluid through the regenerator. This requires work, and this work is not recoverable, so that it is a loss to the cycle. The void volume of the regenerator causes the output mass flow of the regenerator to be less than the input mass flow. The difference is required to "fill" the void volume. In addition, it means that all the mass flow does not flow entirely through the regenerator. Some fraction of that will only traverse a part of the regenerator, and this part will undergo a partial heat exchange process.
In an "ideal" case, the value of hA will be very large when compared to the capacity rate, (mc.sub.p) of the fluid. Here m is the mass fluid flow, and c.sub.p is the heat capacity of the fluid (SI units: m-kg/sec; c.sub.p -Joules/kg.K). In the ideal case the value of the matrix heat capacity, C, must be large when compared to the product .tau.mc.sub.p where .tau. is the "blow" period or a period of time between flow reversals (SI units-sec). In this ideal case, the void volume and pressure drop will be zero. It is impossible to build the ideal regenerator described above since the factors are interrelated. Therefore, all practical regenerators will have pressure drop and void volume. The problem for the regenerator designer is to obtain the necessary values of heat transfer and heat storage, while minimizing the effect of pressure drop and void volume. Previous design efforts have developed a number of different analytical techniques. These techniques must also consider the overall system in which a regenerator is used. However, regardless of the application, certain things are always desirable. These include:
In order to produce very high efficiency regenerators, it is not sufficient simply to provide high thermal capacity material. The material must also be incorporated in an optimum geometry that provides a most effective heat exchange per unit void volume and at the lowest possible pressure drop. Three possible regenerator matrix geometries have been considered and subjected to analysis to determine their relative efficiencies. These regenerators include:
For this analysis to be valid, certain characteristics are required in the perforated plates, in particular, each perforation must have a uniform cross section throughout its length, and the "entry" and "exit" of the perforations must have a sharp right-angle shape. Further considerations are as follows: The "friction factor" and "Stanton number" of tubes with a circular cross section depend on the length-to-diameter ratio (L/D); tubes with a rectangular cross section approach the performance of parallel plates and do not depend on the L/D ratio; and the performance of circular cross section tubes with relatively small L/D approaches that of parallel plates.
Comparisons of heat transfer performance have been made for three study cases:
The results obtained show that:
Predictions of regenerator performance may be made using average temperature values along the entire length of the regenerator. A preferred approach, however, is to section the regenerator and use average values for each section. This requires a knowledge of the temperature gradient along the length of the regenerator, taking into account two main temperature effects: (1) the thermal conductivity of the fluid decreases at lower temperatures so that smaller flow passages are required at low temperatures if effective heat transfer is to be maintained, and (2) the volumetric heat capacity of the matrix decreases at low temperatures, requiring more matrix material.
The regenerator matrix material must be in thermal contact with the fluid in order to be useful. This means that the thermal penetration length, that is, the distance the temperature wave propagates into the matrix, must be long enough that the entire matrix participates in the heat transfer process. For sinusoidal temperature variation, the thermal penetration depth is given by: ##EQU1## where k, .rho., and c.sub.p are the matrix thermal conductivity, density, and specific heat, respectively, and .mu. is the operating frequency. Thus, both a high specific heat and a high thermal conductivity are required to make full use of the matrix heat capacity. This can severely limit the choice of materials.
While the higher efficiency of perforated plates with defined hole geometry is clear, a practical method of fabricating such plates has not been available, particularly at the hole sizes and extent of perforation volume desired for operation of high-performance regenerators at liquid helium temperatures. Hole diameters ranging from 300 microns down to below 1 micron and an open porosity value of 30 to 40 percent of the plate area may be required for specific regenerators, with smaller holes and porosities being required for lower operating temperatures.
Perforated plates for use in various types of heat exchangers are disclosed in prior patents. Hoffman in U.S. Pat. No. 3,273,357, issued Sep. 20, 1966, discloses perforated plates with eight mil diameter holes formed by die cutting or photoetching. U.S. Pat. No. 3,692,099, issued Sep. 19, 1972, to Nesbitt et al. discloses plates with eight mil diameter holes, which are said to be formed by any conventional methods through drilling, punching, etching, or use of sintered matrices of spheres, chips, or wires. U.S. Pat. No. 3,228,460, issued Jan. 11, 1966, to Garwin, shows perforated plates with 15 mil diameter holes but does not disclose how the holes are formed. At the hole sizes of interest for high-performance cryocoolers, that is, from below 1 to 300 microns in diameter, conventional methods as disclosed in these patents are ineffective in that holes produced by these methods do not have a uniformity of shape along their length as required for maximum efficiency. Mechanical methods such as drilling or punching are not practical at these sizes because drills or punches of such sizes are not available and because of the large number of holes required. Photoetching through a mask results in holes of non-uniform shape along their length owing to underetching or other effects that produce curved or inclined, rather than straight, hole walls through the depths of the plate.
Another important factor in the design of high-performance regenerators is the selection of a plate material having optimum thermal properties for the temperature range of operation, in particular, a high specific heat at a selected temperature, consistent with a high thermal conductivity, amenability to fabrication, and reasonable cost. At above 50.degree. K., copper, brass, and 304 stainless steel meet these requirements; at 20 to 50 K, erbium and lead have the highest volumetric heat capacity, and below 20 K, helium and materials with magnetic transitions, in particular GdRh, GdEr.sub.x Rh.sub.1-x, Er.sub.3 Ni and other rare earth alloys have a favorable high heat capacity. Any alloy containing a precious metal such as rhodium would be too expensive. Many of the rare earths are relatively expensive; however, when fabrication costs are included, the cost of some of these materials, in particular Er.sub.3 Ni, would not be prohibitive for high performance applications.
Japanese investigators have performed work on regenerative cryocoolers for use at liquid helium temperatures. (Proceedings of the Sixth International Cryocooler Conference held in Plymouth, Mass., Oct. 25, 1990). This work is directed to regenerators using Er.sub.3 Ni as a heat exchanging material, this compound being selected because of its uniquely high specific heat at temperatures from 3.degree. K. to 20.degree. K. However, it is a brittle intermetallic compound not amenable to fabrication into perforated plates using known methods. The Er.sub.3 Ni in this work was provided in the form of 0.6 mm spheres in a packed bed. Such a geometry does not enable the potentially high performance of this material to be realized. Much better performance would be available if perforated plates of Er.sub.3 Ni with controlled pore geometry would be made.
In addition to providing perforated plates made of selected metals or intermetallic compounds, it is desired to provide a method of fabricating composite plates which would incorporate inclusions of a plate material contained at predetermined locations in a matrix of a first material. The matrix of such a plate would provide the thermal conductivity needed for good heat transfer, and the inclusions would provide the high heat capacity needed for good thermal storage. No method is available for fabricating plates with such a structure. Another desired approach would be to provide perforated plates which include some perforations that would entrap helium and thus take advantage of the high heat capacity of helium.